Download Work and Energy

Work and Energy
1m
4 kg
LAB = mgh
1 kg
1 kg
LAB = 4( m
4 gh) = mgh
LAB = mgl sin α = mgd hd = mgh
Work of a force:
L = F� · �s = F s cos θ = F � s = [N m] = [J] = [Joule]
W = work done by a constant force F� to move a body by
a displacement �s. θ = angle between the vectors F� and �s
(F � is the component of F� parallel to �s).
� (�r) = Fx (�r)î + Fy (�r)ĵ +
Notice: For a non constance force F
Fz (�r)k̂ one needs to calculate the in work, i.e. the work for an
infinitesimal displacement d�r� = dxî + dy ĵ + dz k̂ and sum over
B �
the body path A → B: L = A F
(�r) · d�r
F� �
Constraints (e.g. contact
forces or normal force) do
not contribute to the work!
(by definition they are
perpendicular to the
displacement)
For conservative forces the work does not depends on the path between the initial configuration and the final configuration of the body.
(“the work does not depend on how the work is done”)
Work and Energy
1m
4 kg
LAB = mgh
1 kg
1 kg
LAB = 4( m
4 gh) = mgh
LAB = mgl sin α = mgd hd = mgh
Work of a force:
L = F� · �s = F s cos θ = F s = [N m] = [J] = [Joule]
�
� (�r) = Fx (�r)î + Fy (�r)ĵ +
Notice: For a non constant force F
Fz (�r)k̂ one needs to calculate the in work, i.e. the work for an
infinitesimal displacement d�r� = dxî + dy ĵ + dz k̂ and sum over
B �
the body path A → B: L = A F
(�r) · d�r
In the linear case L =
� xB
xA
F (x) · dx
F� �
Constraints (e.g. contact
forces or normal force) do
not contribute to the work!
(by definition they are
perpendicular to the
displacement)
For conservative forces the work does not depends on the path between the initial configuration and the final configuration of the body.
(“the work does not depend on how the work is done”)
Work and Energy
For conservative forces the work does not depends
on the path between the initial configuration and the
final configuration of the body. (“the work does not
depend on how the work is done”)
Energy: The Energy is the capacity of a body to do a
work, it is expressed in [Joule] (the variation of energy is
the work done by the force acting on the body).
LAB = mgl sin α = mgd hd = mgh
m
m
m/4
h
l
h = l sin α
α
LAB = mgh
LAB = 4( m
4 gh) = mgh
Work and Energy
Gravitational potential energy (on Earth surface)
Ug = mgh
Ug = gravitational potential energy on the Earth surface of a body on mass m
at height y = h relative to a reference position y = 0 (h � rT where rT = Earth
radius)
LAB = mgl sin α = mgd hd = mgh
m
m
m/4
l
h = l sin α
α
LAB = mgh
LAB = 4( m
4 gh) = mgh
Work and Energy
For conservative forces the work does not depends on the path
between the initial configuration and the final configuration of the
body. (“the work does not depend on how the work is done”)
Potential Energy
LAB = −(UB − UA ) = −∆U
� xB
Notice: in general we have ∆U = − xA F (x) · dx = −L
the potential energy of a conservative force is the energy associated to
the position or to the state of a body subject to that force
LAB = mgl sin α = mgd hd = mgh
m
m
m/4
h
l
h = l sin α
α
LAB = mgh
LAB = 4( m
4 gh) = mgh
Work and Energy
Kinetic energy
1
2
K=
mv
2
K = energy of a body m moving at velocity v (= work produce to put the body
at rest).
The total work LAB done on the body that moves from
position A to position B is equal to the variation of its
kinetic energy:
LAB = KB − KA
F� � =
Elastic potential energy
�vA
�
a�
m
�vB
1 2
(for the pendulum replace k → mω 2 )
kx
2
Uk = elastic potential energy of a body attached to a spring displaced of x from
the equilibrium condition; k = elastic constant of the spring.
Uk =
Interpretation of the potential energy graph
A body subject to conservative forces moves toward the minimum of the
corresponding potential energy
U (ti ) + K(ti ) = U (t) + K(t) = U (tf ) + K(tf ) , ∀t
.
.
x(t)
Ug (ti )
Uk (ti )
K(t)
K(t)
UK (t)
Ug (t)
Equilibrium
position
.
h(t)
unstable equilibrium
neutral equilibrium
stable equilibrium
Principle of conservation of Mechanical Energy
In a ISOLATED system the TOTAL energy is conserved
(it is constant in time), but it may transforms from a type
of energy to the other.
For a system on which only conservative forces act
∆K + ∆Ug + ∆Uk + ∆Ualtro = 0
where: ∆K = variation of kinetic energy of the system;
∆Ug = variation of gravitational energy; ∆Uk variation of
elastic energy; ∆Uother variation of other types of conservative potential energy
The energy can not created nor distroied
For dissipative forces (e.g.: friction) the corresponding work
done depends on the particular path and it is not possible to
define a potential. In this case part of the energy of the system is
converted in “internal energy” (e.g.: Heat) of the system and it
can not be directly converted to work.
Gravitational potential energy (general case)
m1 · m2
UG = −G
r
Gravitational potential energy of a point-like particle of
mass m1 at distance r from a second point-like particle
of mass m2 (notice: by convention r → ∞ ⇒ UG → 0).
Escape velocity
�
Em
− GM
RE
�
K + UG = 12 mv 2 +
=0
v is the initial velocity of the projectile of mass m
�
2GME
v=
RE
A Black Hole is an object so
massive that its escape velocity
is begger than the speed of light
Molecular bounding energy
It can be approximated to an
harmonic potential for small
displacement w.r.t. the
equilibrium position.
Atoms have vibrational modes
(hamonic motion) in a analogy
to vibrating springs
Power
Variation of work over time:
Pm
W
=
t
joule
[
≡ W AT T = W ]
second
Pm = mean power of a force is equal to the work done W
divided by the time t taken for doing it.
Another common unit for the Power is the HOURSEPOWER (hp):
1hp = 746W
Notice: the Kilowatt-hour (KW h) is 1kW × 1h = 3.6 ×
106 J = 3.6M J. It denotes an energy (work).
Metabolism
promptness of energy use in leaving beings
Moment of a force w.r.t. a point (a.k.a torque)
�τ = �r × F� → |�τ | = r · F · sin α = F · d = F ⊥ · r
[newton · metro → N · m]
[joule]
If τT OT = 0 the body is at equilibrium w.r.t. rotations
�
τ
F�
�r α
d = r sin α
τ = moment w.r.t. to the point O (fulcrum) of the
force F� applied to the point P . d = lever-arm of the force:
distance between the rect of application of F� from O. �r =
� . α = angle between �r and F�
OP
Wθ = F1⊥ d1 θ = F2⊥ d2 θ → F1⊥ d1 = F2⊥ d2 = cost
�τ1 = �τ2 = �τ = cost
�τ2
�τ1
Momentum of a body
Impulse of force
J� = F� · ∆t ,
[N · s]
J� (mean) impuls of the (constant) force F� applied to a body
for a �time interval ∆t. Total impulse for non-constant forces
t
J� = tif F� (t) · dt
m
[kg · ]
Momentum �q = m · �v ,
s
�q = momentum of a body m and velocity �v .
J� = ∆�q the impulse J� of a force is is the variation of
momentum ∆�q of the body.
q
The second Newton’s law takes the forme F� = d�
dt .
The momentum of an isolated system (F� = 0) is conserved �q = cost.
Elastic (linear) collision between a particle and a target
The elastic collisions between particles are solved by
imposing separately the conservation of the energy and of
the momentum between the conditions of the system before
and after the collision.
m1 v1 = m1 v1� + m2 v2� ,
v1� =
1
1
1
m1 v12 = m1 v1�2 + m2 v2�2
2
2
2
(m1 − m2 )v1
,
m1 + m2
v2� =
before
after
2m1 v1
m1 + m2
Completely inelastic (linear) collision (lineare) with target
In the inelastic collision the particle and target have
the same final velocity. Part of the energy is dissipated
(e.g.: heat). The collision is solved by determining a single
unknown variable by means of the conservation of the total
momentum
m1 v1 = m1 vf� + m2 vf� → vf� =
m1 v 1
m1 + m2